Deriving the Lump-Sum Tax Multiplier

Why a Lump-Sum Tax Change Affects Output

In the Keynesian expenditure model, a change in taxes affects households’ disposable income. When taxes rise, households have less disposable income to spend; when taxes fall, they have more. The resulting change in consumption changes planned expenditure and, through the equilibrium adjustment, changes output. The lump-sum tax multiplier summarizes this relationship as a response of equilibrium output to a change in taxes.

The multiplier is negative: an increase in taxes reduces equilibrium output, while a tax cut increases it. The sign is important, but so is the magnitude. A tax change first changes consumption directly, and the resulting output change then affects income and consumption again. The multiplier captures the total effect in the model, not only the initial consumption response.

Definition: The lump-sum tax multiplier is the change in equilibrium output divided by the change in lump-sum taxes, with other autonomous expenditure held constant. In the basic closed-economy Keynesian model with marginal propensity to consume $c$, it is $\Delta Y/\Delta T=-c/(1-c)$. Here $Y$ is equilibrium output, $T$ is lump-sum tax revenue, and $0<c<1$.

The Model and Its Assumptions

To derive the multiplier, use the basic closed-economy expenditure model. Let consumption be $C=C_0+c(Y-T)$, where $C_0$ is autonomous consumption, $Y-T$ is disposable income, and $c$ is the marginal propensity to consume (MPC). The MPC is the fraction of an additional dollar of disposable income that households spend on consumption. Assume $0<c<1$, so households spend part, but not all, of an additional dollar.

Planned expenditure consists of consumption, investment, and government purchases. In this derivation, investment $I$ and government purchases $G$ are fixed, and there is no foreign trade. The expenditure equilibrium condition is output equals planned expenditure:

$$ Y=C+I+G=C_0+c(Y-T)+I+G. $$

Taxes are lump-sum: $T$ is a fixed amount in the model, not a rate that rises automatically with income. To interpret the multiplier as a response to a tax change, also hold $C_0$, $I$, $G$, and $c$ constant. The calculation describes the model’s equilibrium adjustment; it does not by itself account for changes in prices, interest rates, productive capacity, or policy responses.

A change in taxes is written $\Delta T$. By convention, a tax increase has $\Delta T>0$, while a tax cut has $\Delta T<0$. The corresponding change in equilibrium output is $\Delta Y$. Keeping these changes distinct prevents a frequent sign error: the multiplier is negative, but a tax cut is itself a negative change, so its effect on output is positive.

Deriving the Multiplier

Compare two equilibria in the same model: one before a tax change and one after it. Since autonomous consumption, investment, government purchases, and the MPC are unchanged, subtracting the first equilibrium condition from the second eliminates those unchanged terms. The change in consumption is $c(\Delta Y-\Delta T)$, because disposable income changes by the change in output minus the change in taxes.

Write the change in equilibrium expenditure.
With investment and government purchases fixed, the change in output must equal the change in consumption: $\Delta Y=\Delta C$.

Express the change in consumption.
Since $C=C_0+c(Y-T)$ and $C_0$ and $c$ are fixed, $\Delta C=c(\Delta Y-\Delta T)$.

Solve for the output response.
Substitute the consumption change into equilibrium and collect the $\Delta Y$ terms.

\begin{aligned} \Delta Y&=c(\Delta Y-\Delta T),\\ \Delta Y&=c\Delta Y-c\Delta T,\\ (1-c)\Delta Y&=-c\Delta T,\\ \frac{\Delta Y}{\Delta T}&=-\frac{c}{1-c}. \end{aligned}

The denominator $1-c$ is positive because the model assumes $c<1$, so the multiplier is negative. For a tax increase, $\Delta T>0$, and therefore $\Delta Y<0$. For a tax cut, $\Delta T<0$, and therefore $\Delta Y>0$. The sign follows directly from the equilibrium equation rather than from a memorized rule.

There is also a useful spending-round interpretation of the derivation. A tax increase of $\Delta T$ first reduces disposable income by that amount, lowering consumption by $c\Delta T$. This initial consumption reduction lowers output. The lower output then reduces income and consumption again, and each subsequent round is $c$ times the preceding round. For $0<c<1$, the geometric series converges:

\Delta Y=-c\Delta T-c^2\Delta T-c^3\Delta T-\cdots =-c\Delta T\left(\frac{1}{1-c}\right) =-\frac{c}{1-c}\Delta T.

This gives the same result and explains why the total output response is larger in magnitude than the initial consumption change: the initial change triggers further rounds of induced consumption. The formula applies to a change in the tax amount, not to the entire tax level by itself.

Using the Multiplier in Calculations

For a numerical problem, identify the MPC, establish whether taxes rise or fall, and keep the units of the tax change and output change consistent. Then calculate the multiplier and multiply it by $\Delta T$. If the problem provides a consumption function and asks for output levels as well as the change, use the equilibrium equation before and after the tax change to verify the result.

Worked Example: A Tax Increase in a Local Economy

A hypothetical closed economy has an MPC of $0.75$. The government raises lump-sum taxes by $\$40$ million. Assume investment, government purchases, autonomous consumption, and the MPC do not change. Find the change in equilibrium output and verify the direction of the effect.

Method. Apply the lump-sum tax multiplier. The tax change is positive because taxes increase: $\Delta T=+\$40$ million. The multiplier is

-\frac{c}{1-c} =-\frac{0.75}{1-0.75} =-\frac{0.75}{0.25} =-3.

Now multiply by the tax change:

\Delta Y=(-3)(+\$40\text{ million}) =-\$120\text{ million}.

Thus equilibrium output falls by $\$120$ million. The sign is consistent with the mechanism: the tax increase initially reduces consumption, and the induced reductions in output and consumption extend the total effect. As a check on the direction, $\Delta T>0$ and the multiplier is negative, so $\Delta Y<0$. The common mistake avoided here is reporting a positive output change merely because the tax change was entered as a positive amount.

Worked Example: A Tax Cut and the Sign of the Change

In a second hypothetical closed economy, the MPC is $0.60$. The government cuts lump-sum taxes by $\$24$ million. All other autonomous expenditure and model parameters are fixed. Calculate the output change and show how the first consumption response relates to the total.

A tax cut is a negative change in taxes, so $\Delta T=-\$24$ million. First calculate the multiplier:

-\frac{c}{1-c} =-\frac{0.60}{1-0.60} =-\frac{0.60}{0.40} =-1.5.

The total output change is

\Delta Y=(-1.5)(-\$24\text{ million}) =+\$36\text{ million}.

The initial increase in disposable income is $\$24$ million, so the first increase in consumption is $0.60(\$24\text{ million})=\$14.4$ million. Further rounds are $0.60(\$14.4\text{ million})=\$8.64$ million, then $0.60(\$8.64\text{ million})=\$5.184$ million, and so on. The total is $14.4+8.64+5.184+\cdots=14.4/(1-0.60)=\$36$ million. This agrees with the multiplier calculation. The common mistake avoided is treating the tax cut as a positive $\Delta T$; doing so would reverse the sign incorrectly.

Worked Example: Checking the Result From Equilibrium Output Levels

A hypothetical closed economy has consumption $C=50+0.80(Y-T)$, with output and all expenditure amounts measured in thousands of dollars. Investment is $I=30$, government purchases are $G=20$, and lump-sum taxes initially equal $T=25$. The government then raises taxes to $T=35$. Find equilibrium output before and after the change, and verify the multiplier result.

Initial equilibrium. Substitute the consumption function and initial taxes into $Y=C+I+G$:

\begin{aligned} Y&=50+0.80(Y-25)+30+20\\ &=50+0.80Y-20+50\\ &=0.80Y+80. \end{aligned}

Subtract $0.80Y$ from both sides: $0.20Y=80$. Therefore $Y=80/0.20=400$ thousand dollars.

New equilibrium. With taxes at $35$, the equilibrium equation is

\begin{aligned} Y&=50+0.80(Y-35)+30+20\\ &=50+0.80Y-28+50\\ &=0.80Y+72. \end{aligned}

Subtract $0.80Y$: $0.20Y=72$, so the new equilibrium is $Y=72/0.20=360$ thousand dollars. The change is $360-400=-40$ thousand dollars.

Multiplier check. Here $\Delta T=35-25=10$ thousand dollars, and the multiplier is $-0.80/(1-0.80)=-0.80/0.20=-4$. Thus $\Delta Y=(-4)(10)=-40$ thousand dollars, matching the difference between the two equilibrium levels. The common mistake avoided is using the new tax level of $35$ as though it were the tax change; the multiplier applies to the increase of $10$, not to the full tax amount.

Interpretation and Common Traps

The magnitude $c/(1-c)$ describes how many units of output change, in absolute value, per unit change in lump-sum taxes under the stated assumptions. For example, an MPC of $0.75$ gives a multiplier of $-3$: each additional dollar of lump-sum taxes is associated with a three-dollar reduction in equilibrium output in this model. This is a model result, not a claim that every observed tax change in a real economy produces that exact response.

The formula depends on the assumptions used to derive it. If taxes change automatically with income rather than remaining lump-sum, the change in disposable income is not simply $\Delta Y-\Delta T$ with a fixed tax amount. Likewise, changes in imports, interest-sensitive investment, prices, or policy responses would require a different or expanded model. Do not use the basic expression without checking that the problem’s setup matches it.

Wrong sign for a tax cut: Writing $\Delta T=+24$ for a $\$24$ tax cut makes the calculated output change negative. The correction is $\Delta T=-24$; multiplying by the negative multiplier then gives a positive $\Delta Y$.
Using the wrong denominator: The multiplier is $-c/(1-c)$. Since $1-c$ is the share of additional disposable income not consumed, it is the denominator obtained when collecting the output terms in equilibrium.
Using the tax level instead of its change: If taxes rise from $25$ to $35$, use $\Delta T=10$, not $35$. The multiplier relates changes to changes.
Stopping at the initial consumption response: The initial effect of a tax change on consumption is $-c\Delta T$, not the total change in equilibrium output. Subsequent induced-consumption rounds are included in the multiplier.
Ignoring units: If the tax change is stated in millions of dollars, report the output change in millions; if amounts are in thousands, keep the calculation in thousands.

Key takeaway: In the basic closed-economy model with fixed lump-sum taxes, $0<c<1$, and other autonomous expenditure held constant, $\Delta Y/\Delta T=-c/(1-c)$. A tax increase lowers equilibrium output; a tax cut raises it. Always use the signed tax change, not the tax level.

Check Your Understanding

For each problem, use the closed-economy lump-sum tax model and keep the sign of the tax change explicit.

A hypothetical economy has an MPC of $0.70$. Lump-sum taxes rise by $\$10$ million, with other autonomous expenditure fixed. What is the resulting change in equilibrium output? (Answer: $-\$23.33$ million.)
A closed economy has an MPC of $0.80$. The government cuts lump-sum taxes by $\$15$ million. Calculate the equilibrium output change and state whether it rises or falls. (Answer: $+\$60$ million; output rises.)
In a hypothetical economy, $C=90+0.75(Y-T)$, $I=40$, $G=30$, and initially $T=20$, with all amounts in thousands of dollars. Calculate initial equilibrium output. If taxes rise to $28$, calculate the new equilibrium and the change in output. (Answer: Initial output $=\$580$ thousand; new output $=\$556$ thousand; change $=-\$24$ thousand.)
An economy with an MPC of $0.60$ experiences a lump-sum tax increase of $\$25$ million. Calculate the tax multiplier and the output change. (Answer: Multiplier $=-1.5$; output change $=-\$37.5$ million.)
A tax increase of $\$12$ million is associated, in the basic model, with an equilibrium output decrease of $\$48$ million. What MPC is implied? (Answer: $c=0.80$.)

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